UNIQUE ERGODICITY OF TRANSLATION FLOWS

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1 UNIQUE ERGODICITY OF TRANSLATION FLOWS YITWAH CHEUNG AND ALEX ESKIN Abstract. This preliminary report contains a sketch of the proof of the following result: a slowly divergent Teichmüller geodesic satisfying a certain logarithmic law is determined by a uniquely ergodic measured foliation. 1. Introduction Let (X, ω) be a translation surface. This means X is a closed Riemann surface and ω a holomorphic 1-form on X. The line element ω induces a flat metric on X which has cone-type singularites at the zeroes of ω where the cone angle is a integral multiple of 2π. A saddle connection in X is a geodesic segment with respect to the flat metric that joins a pair of zeroes of ω without passing through one in its interior. Our main result is a new criterion for the unique ergodicity of the vertical foliation F v, defined by Re ω = 0. Teichmüller geodesics. The complex structure of X is uniquely determined by the atlas {(U α, ϕ α )} of natural parameters away from the zeroes of ω specified by dϕ α = ω. The evolution of X under the Teichmüller flow is the family of Riemann surfaces X t obtained by postcomposing the charts with the R-linear map z e t/2 Re z+ie t/2 Im z. It defines a unit-speed geodesic with respect to the Teichmüller metric on the moduli space of compact Riemann surfaces normalised so that Teichmüller disks have constant curvature 1. The Teichmüller map f t : X X t takes rectangles to rectangles of the same area, stretching in the horizontal direction and contracting in the vertical direction by a factor of e t/2. By a rectangle in X we mean a holomorphic map a product of intervals in C such that ω pulls back to dz. All rectangles are assumed to have horizontal and vertical edges. Date: September 17, Mathematics Subject Classification. 32G15, 30F30, 30F60, 37A25. Key words and phrases. Abelian differentials, Teichmüller geodesics, Boshernitzan s criterion, Masur s criterion, slow divergence. The first author acknowledges the Fields Institute for financial support to attend the Workshop on Partially hyperbolic dynamics, laminations and Teichmüller flow, Toronto, Jan. 5-9,

2 2 YITWAH CHEUNG AND ALEX ESKIN Let l(x t ) denote the length of the shortest saddle connection. Let d(t) = 2 log l(x t ). In this note we give a sketch of the proof of the following result. Theorem 1.1. There is an ε > 0 such that if d(t) < ε log t + C for some C > 0 and for all t > 0, then F v is uniquely ergodic. The hypothesis of Theorem 1.1 can be formulated in terms of the flat metric on X without appealing to the forward evolution of the surface. Let h(γ) and v(γ) denote the horizontal and vertical components of a saddle connection γ, which are defined by h(α) = Re ω and v(α) = Im ω. γ It is not hard to show that the following statements are equivalent. (a) There is a C > 0 such that for all t > 0, d(t) < ε logt + C. (b) There is a c > 0 such that for all t > 0, l(x t ) > c/t ε/2. (c) There are constants c > 0 and h 0 > 0 such that for all saddle connections γ satisfying h(γ) < h 0, (1) h(γ) > c v(γ)(log v(γ)) ε. For any p > 1/2 there are translation surfaces with nonergodic F v whose Teichmüller geodesic X t satisfies the sublinear slow rate of divergence d(t) Ct p. See [Ch2]. Our main result asserts a logarithmic slow rate of divergence is enough to ensure unique ergodicity of F v. To illustrate our techniques, we give a new proof of the following well-known result, which is a special case of Masur s criterion for unique ergodicity [Ma] which asserts that F v is uniquely ergodic as soon as X t has an accumulation point in the moduli space of compact Riemann surfaces. Theorem 1.2. If l(x t ) does not approach zero as t then F v is uniquely ergodic. Remark. Theorem 1.2 can also be proven using Boshernitzan s criterion for interval exchange transformations, [Ve1], [Ve2]. 1 On the other hand, if (X, ω) is the suspension of a minimal interval exchange transformation satisfying Boshernitzan s criterion one can show that l(x t ) does not tend to zero. It is not hard to give nonergodic examples of F v where the lim inf of d(t) goes to infinity arbitrarily slowly. 1 It should be pointed out that Masur s result is misquoted in [Ve2]. γ

3 UNIQUE ERGODICITY OF TRANSLATION FLOWS 3 Theorem 1.3. Given any function r(t) as t there exists a Teichmüller geodesic X t with nonergodic F v and lim inf d(t)/r(t) < 1. Outline. After introducing some notation and terminology in 2 we prove Theorem 1.2 in 3. Then we explain what modifications are necessary to obtain a proof of Theorem 1.1. A proof of Theorem 1.3 is included in 5 and can be read independent of the other sections. 2. Generic Points If ν is a (normalised) ergodic invariant measure transverse to the vertical foliation F v then for any horizontal arc I there is a full ν- measure set of points x X satisfying (2) lim # I L x L x = ν(i) as L x where L x represents a vertical segment having x as an endpoint. Given I, the set E(I) of points satisfying (2) for some ergodic invariant ν has full Lebesgue measure. We refer to the elements of E(I) as generic points and the limit in (2) as the ergodic average determined by x. Convention. When passing to a subsequence t n along the Teichmüller geodesic X t we shall suppress the double subscript notation and write X n instead of X tn. Similarly, we write f n instead of f tn. Lemma 2.1. Let x, y E(I) and suppose there is a sequence t n such that for every n the images of x and y under f n lie in a rectangle R n X n and the sequence of heights h n satisfy lim h n e tn/2 =. Then x and y determine the same ergodic averages. Proof. One can reduce to the case where f n (x) and f n (y) lie at the corners of R n. Let n (resp. n + ) be the number of times the left (resp. right) edge of fn 1R n intersects I. Observe that n and n + differ by at most one so that since h n e tn/2, the ergodic averages for x and y approach the same limit. Ergodic averages taken as T are determined by fixed fraction of the tail: for any given λ (0, 1) 1 T T 0 f c implies 1 (1 λ)t T λt f c. This elementary observation is the motivation behind the following. Definition 2.2. A point x is K-visible from a rectangle R if the vertical distance from x to R is at most K times the height of R. We have the following generalisation of Lemma 2.1.

4 4 YITWAH CHEUNG AND ALEX ESKIN Lemma 2.3. If x, y E(I), t n and K > 0 are such that for every n the images of x and y under f n are K-visible from some rectangle whose height h n satisfies h n e tn/2, then x and y determine the same ergodic averages. 3. Networks To prove unique ergodicity we shall show that all ergodic averages converge to the same limit. The ideas in this section were motivated by the proof of Theorem 1.1 in [Ma]. Definition 3.1. We say two points are K-reachable from each other if there is a rectangle R from which both are K-visible. We also say two sets are K-reachable from each other if every point of one is K- reachable from every point of the other. Definition 3.2. Given a collection A of subsets of X, we define an undirected graph Γ K (A) whose vertex set is A and whose edge relation is given by K-reachability. A subset Y X is said to be K-fully covered by A if every y Y is K-reachable from some element of A. We say A is a K-network if Γ K (A) is connected and X is K-fully covered by A. To prove Theorem 1.2 we first need a proposition. Proposition 3.3. If K > 0, N > 0, δ > 0 and t n are such that for all n, there exists a K-network A n in X n consisting of at most N squares, each having measure at least δ, then F v is uniquely ergodic. Proof. Suppose F v is not uniquely ergodic. Then we can find a distinct pair of ergodic invariant measures ν 0 and ν 1 and a horizontal arc I such that ν 0 (I) ν 1 (I). We construct a finite set of generic points as follows. By allowing repetition, we may assume each A n contains exactly N squares, which shall be enumerated by A(n, i), i = 1,...,N. Let A 1 X be the set of points whose image under f n belongs to A(n, 1) for infinitely many n. Note that A 1 has measure at least δ because it is a descending intersection of sets of measure at least δ. Hence, A 1 contains a generic point; call it x 1. By passing to a subsequence we can assume the image of x 1 lies in A(n, 1) for all n. By a similar process we can find a generic point x 2 whose image belongs to A(n, 2) for all n. When passing to the subsequence, the generic point x 1 retains the property that its image lies in A(n, 1) for all n. Continuing in this manner, we obtain a finite set F consisting of N generic points x i with the property that the image of x i under f n belongs to A(n, i) for all n and i.

5 UNIQUE ERGODICITY OF TRANSLATION FLOWS 5 Given a nonempty proper subset F F we can always find a pair of points x F and y F \F such that f n (x) and f n (y) are K-reachable from each other for infinitely many n. This follows from the fact that Γ K (A n ) is connected. By Lemma 2.3, the points x and y determine the same ergodic averages for any horizontal arc I. Since F is finite, the same holds for any pair of points in F. Now let z j be a generic point whose ergodic average is ν j (I), for j = 0, 1. Since X n is K-fully covered by A n, z j will have the same ergodic average as some point in F, which contradicts ν 0 (I) ν 1 (I). Therefore, F v must be uniquely ergodic. Proof of Theorem 1.2. Fix δ > 0 such that every saddle connection in X t has length greater than 2δ along some subsequence t n. Note that any immersed square of side at most δ is embedded so that its Lebesgue measure is equal to its area. Consider the Delaunay triangulation of X n. (See [MS].) It has 2ν triangles and 3ν edges where ν depends only on the genus and the number of singularities. Each triangle is contained in an immersed circumscribing disk of radius at least δ. It is not hard to see that every point in the disk is 1-reachable from the central square of side δ located at the circumcenter. Let A n = {A i } 2ν i=1 be the collection of central squares and note that X n is 1-fully covered by A n. Let γ j be a Delaunay edge. It lies on the boundary of exactly two Delaunay triangles, each having circumcenter on the perpendicular bisector of γ j. Pick any square of side δ centered at some point on γ j and divide it into 9 nonoverlapping squares of side δ/3. Two of the four corner squares are disjoint from γ j. Let A j (resp. A j) be the corner square that is disjoint from γ j and lies on the same side of one of the circumcenters (resp. the other circumcenter), possibly with A j = A j. One verifies easily that by adding these squares to the collection A n we obtain a 1-network of at most 8ν squares, each with area at least δ 2 /9. The theorem now follows from Proposition Buffered Squares Our interest lies in the case where l(x t ) 0 as t. To prove of Theorem 1.1 we shall need an analog of Proposition 3.3 that applies to a continuous family of networks A t whose squares have dimensions going to zero. We also need to show that the slow rate of divergence gives us some control on the rate at which the small squares approach zero. Recall that in the proof of Theorem 1.2 it was essential that the measure of the squares in the networks stay bounded away from zero.

6 6 YITWAH CHEUNG AND ALEX ESKIN This allowed us to find generic points that persist in the squares of the networks along a subsequence t n. If the area of the squares tend to zero slowly enough, one can still expect to find persistent generic points, with the help of the following result from probability theory. Lemma 4.1. (Paley-Zygmund [PZ]) If A n be a sequence of measureable subsets of a probability space satisfying (i) A n A m K A n A m for all m > n, and (ii) A n = then A n i.o. 1/K. Definition 4.2. We say a rectangle is δ-buffered if it can be extended in the vertical direction to a larger rectangle that overlaps itself at most once and has area at least δ. Here, we only require the product of the dimensions of the larger rectangle be at least δ, which obviously holds if its Lebesgue measure exceeds δ. Proposition 4.3. Suppose that for every t > 0 we have a δ-buffered square A t embedded in X t with side σ t > c/t ε/2, 0 < ε 1. Then there exists t n such that conditions (i) and (ii) of Lemma 4.1 are satisfied by the sequence fn 1 A n of rectangles in X for some K depending only on c and δ. Proof. Let (t n ) be any sequence satisfying the recurrence relation t n+1 = t n + ε log(t n+1 ) t 0 > 1. Note that the function y = y(x) = x ε log x is increasing for x > ε and has inverse x = x(y) is increasing for y > 1, from which it follows that (t n ) is increasing. We have σ m e tm tn > (ctm ε/2 )tε n+1 tε/2 m > ctε/2 n. Let B n A n be the rectangle in X n that has the same width as A n and area at least δ. Let α(b n ) denote the product of the dimensions of B n. Since B n overlaps itself at most once, α(b n ) 2 B n 2. Therefore, the height of B n is < 2/σ n < (2/c)t ε/2 n, which is less than 2/c 2 times the height of the rectangle A m = f n fm 1 A m, by the choice of t n. Let A m be the smallest rectangle containing A m that has horizontal edges disjoint from the interior of B n. Its height is at most 1 + 4/c 2 times that of A m. For each component I of A n A m there is a corresponding component J of B n A m (see Figure 1) so that A n A m = I α(a n) J δ 1 A n α(a α(b n ) m) < 4 + c2 A δc 2 n A m

7 UNIQUE ERGODICITY OF TRANSLATION FLOWS 7 giving (i). Choose t 0 large enough so that t n+1 < 2t n for all n and suppose that for some C > 0 and n > 1 we have t n < Cn log n log log n. Then t n+1 < t n + log t n+1 < t n + log t n + log 2 < Cn log n log log n + log n + log log n + log log log n + log 2C < Cn log n log log n + log n log log n for n 1 < C(n + 1) log(n + 1) log log(n + 1) so that t n O(n lognlog log n). Since An = σ 2 n > c 2 and ε 1, (ii) follows. t ε n There is a simple procedure for constructing buffered squares. Take any saddle connection γ and decompose the surface into a finite number of vertical strips, each of which is a parallelogram with a zero on each of its vertical edges and the remaining pair of edges contained in γ. The number of vertical strips is ν = 2g 1 + r where g is the genus of X and r the number of zeroes of ω. Since one of the strips has area 1/ν, any rectangle containing the strip serves as a 1/ν-buffer for any square of the same width contained in it. See Figure 2. The condition (1) prevents the slopes of saddle connections from being too close to vertical. This allows for some control on the widths of vertical strips. Proposition 4.4. There is a constant c > 0 depending only on the constants of the logarithmic rate such that for any t > 1 and any saddle connection γ in X t of length at most 1 there is a vertical strip with a pair of edges contained in γ, having area 1/ν and width c/t νε. By an argument similar to the proof of Theorem 1.2 one constructs a network of small buffered squares in X t whose areas tend to zero no faster than 1/t 2νε where ε is the coefficient of the logarithmic slow rate. Proposition 4.4 is used repeatedly on the edges of the Delaunay triangulation to construct the individual buffered squares of the network. (The main technical obstruction is that some of the edges of Delaunay triangulation may be too long so that we cannot apply Proposition 4.4; however, long edges are well understood they have to cross long thin cylinders.) Invoking Proposition 4.3 we pass to a subsequence where the hypotheses of the lemma of Paley-Zygmund hold. An argument similar to the proof of Proposition 3.3 leads to the conclusion that F v is uniquely ergodic. Indeed, by the argument outlined above one can

8 8 YITWAH CHEUNG AND ALEX ESKIN B n A m A n A m Figure 1. For purposes of illustration, the rectangles are represented by their images in X t where t is the unique time when B n maps to a square under the composition f t fn 1 of Teichmüller maps. take ε = 1/(2ν) in Theorem 1.1. A complete proof of Theorem 1.1 will appear elsewhere. 5. Arbitrarily slow liminf divergence The examples with nonergodic vertical foliation are exhibited in a well-known family of branched double covers of tori. For background regarding the construction, we refer the reader to [MT].

9 UNIQUE ERGODICITY OF TRANSLATION FLOWS 9 Figure 2. Any rectangle contained the strip with largest area serves as a buffer for any square of the same width contained in it. Proof of Theorem 1.3. Let X be the connected sum of T 2 = (C/(Z + iz), dz) with itself along a horizontal arc I of irrational length λ (0, 1). Let W = {λ + 2m + 2ni : m, n Z, n > 0} and note that each element in W is a vector with irrational slope. The holonomy of a saddle connection γ in X joining the two endpoints of the slit I is of the form λ + m + ni for some m, n Z and since λ is irrational, every such number is the holonomy of some saddle connection in X provided n 0. It is convenient to think of elements in W as saddle connections. By concatenating γ with its image under the hyperelliptic involution, which interchanges the endpoints of I, we obtain a simple closed curve Γ. The complement of Γ has two connected components if and only if m and n are both even integers; thus, every element of W determines a partition of X into two sets of equal measure. For more details, see [Ch1].

10 10 YITWAH CHEUNG AND ALEX ESKIN Fix a summable series of positive terms a j <. Let w 0 W. Given w j W we can find an arbitrarily long v j Z + iz such that w j v j < a j. (Here, z w := Im ( zw).) We set w j+1 = w j + 2v j and note that w j+1 W and that w j w j+1 < 2a j. The symmetric difference between the partitions determined by w j and w j+1 is a union of parallelograms whose total area is bounded above by the cross product. Applying Masur-Smillie s criterion for non-ergodicity [MS], we see that the directions of w j converge to a nonergodic direction θ. Rotate so that θ is vertical and let X t be the evolution of X under the Teichmüller flow. Let wj t be the image of w j in X t. Let t j be the unique time t = t j when wj t and wt j+1 have the same length. By choosing v j so that the vertical (resp. horizontal) component of w j+1 is at least twice (resp. at most half) that of w j we can ensure that the angle between wj t and wt j+1 is at least cos 1 (4/5), the angle between the vectors 1+2i and 2+i. Hence, l(x t ) 2 wj w t j+1 t (5/3) w j w j+1 so that d(t j ) log w j w j+1 + C. By choosing v j long enough we can make d(t j )/r(t j ) < 1/2, which completes the proof. References [Ch1] Cheung, Yitwah. Hausdorff dimension of the set of nonergodic directions. Ann. of Math., 158 (2003), [Ch2] Cheung, Yitwah. Slowly divergent trajectories in moduli space, Conform. Geom. Dyn., 8 (2004), [Ma] Masur, Howard. Hausdorff dimension of the set of nonergodic foliations of [MS] a quadratic differential Duke Math. Journal, 66 (1992), Masur, Howard and Smillie, John. Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math., 134 (1991), [MT] Masur, Howard and Tabacknikov, Serge. Rational billiards and flat structures, in Handbook of Dynamical Systems 1A (2002), [PZ] Paley, R.E.A.C and Zygmund, A. Proc. Camb. phil. Soc. 26 (1930), , and 28 (1932), [Ve1] Veech, William A. Boshernitzan s criterion for unique ergodicity of an interval exchange transformation. Ergodic Theory Dynam. Systems, 7 (1987), [Ve2] Veech, William A. Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form. Ergodic Theory Dynam. Systems, 19 (1999), San Francisco State University, San Francisco, California address: cheung@math.sfsu.edu University of Chicago, Chicago, Illinois address: eskin@math.uchicago.edu